Answer:
(A) considering the charge "q" evenly distributed, applying the technique of charge integration for finite charges, you obtain the expression for the potential along any point in the Z-axis:

With  been the vacuum permittivity
 been the vacuum permittivity 
(B) The expression for the magnitude of the E(z) electric field along the Z-axis is:

Explanation:
(A) Considering a uniform linear density  on the ring, then:
 on the ring, then:
 (1)⇒
 (1)⇒ (2)⇒
(2)⇒ (3)
(3)
Applying the technique of charge integration for finite charges:
 (4)
(4)
Been r' the distance between the charge and the observation point and a, b limits of integration of the charge. In this case a=2π and b=0.
Using cylindrical coordinates, the distance between a point of the Z-axis and a point of a ring with R radius is:
 (5)
(5)
Using the expressions (1),(4) and (5) you obtain:

Integrating results:
 (S_a)
   (S_a)
(B) For the expression of the magnitude of the field E(z), is important to remember:
 (6)
 (6)
But in this case you only work in the z variable, soo the expression (6) can be rewritten as:
 (7)
 (7)
Using expression (7) and (S_a), you get the expression of the magnitude of the field E(z):
 (S_b)
 (S_b)